# 经济代写|计量经济学代写Econometrics代考|ECON2271

We remarked in the introduction to this chapter that the Gauss-Newton regression is not generally applicable to models estimated by maximum likelihood. In view of the extreme usefulness of the GNR for computing test statistics in the context of nonlinear regression models, it is of much interest to see if other artificial regressions with similar properties are available in the context of models estimated by maximum likelihood.

One preliminary and obvious remark: No regression, artificial or otherwise, is needed to implement the LR test. Since any package capable of producing ML estimates will certainly also produce the maximized loglikelihood function, there can be no obstacle to performing an LR test unless there is some difficulty in estimating either the restricted or the unrestricted model. In many cases, there is no such difficulty, and then the LR test is almost always the procedure of choice. However, there are occasions when one of the two models is much easier to estimate than the other, and then one would wish to use either the LM or the Wald test to avoid the more difficult estimation. Another possibility is that the alternative hypothesis may be implicit rather than being associated with a well-defined parametrized model that includes the null hypothesis as a special case. We have seen in the context of the GNR that many diagnostic tests fall into this category. When the alternative hypothesis is implicit, one would almost always wish to use an LM test.

In the regression context, the GNR provides a means of computing test statistics based on the LM principle. In point of fact, as we saw in Section 6.7, it can be used to compute test statistics based on any root- $n$ consistent estimates. We will now introduce a new artificial regression, called the outerproduct-of-the-gradient regression, or the OPG regression for short, which can be used with any model estimated by maximum likelihood. The OPG regression was first used as a means of computing test statistics by Godfrey and Wickens (1981). This artificial regression, which is very easy indeed to set up for most models estimated by maximum likelihood, can be used for the same purposes as the GNR: verification of first-order conditions for the maximization of the loglikelihood function, covariance matrix estimation, one-step efficient estimation, and, of greatest immediate interest, the computation of test statistics.

The three classical tests, as the word “classical” implies, have a long history and have generated a great deal of literature; see Engle (1984) and Godfrey (1988) for references. In this chapter, we have tried to emphasize the common aspects of tests underlying the very considerable diversity of testing procedures and to emphasize the geometrical interpretation of the tests. A simpler discussion of the geometry of the classical tests may be found in Buse (1982). We have pointed out that there is a common asymptotic random variable to which all the classical test statistics tend as the sample size tends to infinity and that the distribution of this asymptotic random variable is chi-squared, central if the null hypothesis under test is true, and noncentral otherwise. The actual noncentrality parameter is a function of the drifting DGP considered as a model of the various possibilities that exist in the neighborhood of the null hypothesis. Because the mathematics involved is not elementary, we did not discuss the details of how this noncentrality parameter may be derived, but the intuition is essentially the same as for the case of nonlinear regression models discussed in Section 12.4.

The asymptotic properties of the classical tests under DGPs other than those satisfying the null hypothesis is studied in a well-known article of Gallant and Holly (1980) as well as in the survey article of Engle (1984). In these articles, only drifting DGPs that satisfied the alternative hypothesis were taken into account. The Gallant and Holly article provoked a substantial amount of further research. One landmark of the literature in which this research is reported is a paper by Burguete, Gallant, and Souza (1982), in which an ambitious project of unification of a wide variety of asymptotic methods is undertaken. Here, for the first time, drifting DGPs were considered which, although in the neighborhood of the null hypothesis, satisfied neither the null nor the alternative hypothesis. Subsequently, Newey (1985a) and Tauchen (1985) continued the investigation of this approach and were led to propose new tests and still more testing procedures (see Chapter 16). Our own paper (Davidson and MacKinnon, 1987) pursued the study of general local DGPs and was among the first to try to set the theory of hypothesis testing in a geometrical framework in such a way that “neighborhoods” of a null hypothesis could be formally defined and mentally visualized. The geometrical approach had been gaining favor with econometricians and, more particularly, statisticians for some time before this and had led to the syntheses found in Amari (1985) and Barndorff-Nielsen, Cox, and Reid (1986); see the survey article by Kass (1989). We should warn readers, however, that the last few references cited use mathematics that is far from elementary.

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